Structures of Adjoint-Stable Algebras over Factorizable Hopf Algebras

For a quasi-triangular Hopf algebra $\left( H,R\right)$, there is a notion of transmuted braided group $H_{R}$ of $H$ introduced by Majid. The transmuted braided group $H_{R}$ is a Hopf algebra in the braided category $_{H}\mathcal{M}$. The $R$-adjoint-stable algebra associated with any simple left $H_{R}$-comodule is defined by the authors, and is used to characterize the structure of all irreducible Yetter-Drinfeld modules in ${}_{H}^{H} \mathcal{YD}$. In this note, we prove for a semisimple factorizable Hopf algebra $\left( H,R\right)$ that any simple subcoalgebra of $H_R$ is $H$-stable and the $R$-adjoint-stable algebra for any simple left $H_R$-comodule is anti-isomorphic to $H$. As an application, we characterize all irreducible Yetter-Drinfeld modules.